Transcript 下載/瀏覽

Other derivation of the least squares estimates
According to the principle of least square, we must determine the quantity of b0 and b1 such
n
that
D  (y  b  b x )
i
i 1
0
1
2
is minimized. First we write
i
( y  b  b x )  ( y  y )  b ( x  x)  ( y  b  b x)
i
0
1 i
i
1
i
0
1
Squaring both sides we then obtain
( y  b  b x )  ( y  y )  b (0x  x)  ( y  b  b x)
2
i
0
1
2
2
i
i
1
2
i
0
2
0
1
 2b ( y  y )( x  x)  2b ( x  x)( y  b  b x)  2( y  y )( y  b  b x)
1
i
i
We now sum both sides over
1
i
0
1
And  ( y  yi )  0 Hence we have
i 1
0
1
i  1,2,..., n and note that the last two terms in the right hand
side of the formular disappear after summation, because
n
i
n
 (x  x )  0
i 1
i